AEA Papers and Proceedings 2021, 111: 544–548
https://doi.org/10.1257/pandp.20211053

Learning and Corruption on Monitoring Chains†
By Bruno Strulovici*
Many institutions rely on specific agents, or“monitors,” for their enforcement. These monitors are subject to incentive problems such as
shirking—emphasized by the inspection games
literature (Dresher 1962, Maschler 1966)—fabrication, and corruption (Becker and Stigler
1974).
This issue is fundamental for the enforcementof legal and political institutions as well as fornumerous other institutions, such as trade institutions (Milgrom, North, and Weingast 1990),
financial markets, and agencies in charge of taxation and auditing.
When monitors cannot merely be trusted tobehave ethically, they must also be monitored,
their monitors must be monitored, and so on.
This issue has been recognized by Hurwicz(2007), who proposed a cyclical incentive structure: ​B​monitors A​ ​, ​C​monitors ​B​’s monitoringof ​A​, ​A​monitors ​C​’s monitoring of ​B​’s monitoring of ​A​.1
However, this structure seems vulnerable
to collusion, not least because monitors exert
indirect control over their own monitors. In
practice, monitors are rarely monitored by theirown target, even indirectly. However, collusion
between enforcers is a real and well-documented
concern.2
This paper studies the effect of collusion along
monitoring chains, along which each monitor ismonitored by a new monitor. The main result
is that when (i) monitors are devoid of ethicalmotives and have quasilinear preferences and(ii) any two consecutive monitors in the chain
can make ­Pareto-improving corruptive arrangements, truthful, ­collusion-free monitoring is
impossible unless rewards and/or punishments
are unbounded.
In these monitoring chains, even a local
form of collusion suffices to derail monitoring.
This result underlines the importance of ethicalbehavior for the enforcement of institutions.
In a separate paper (Strulovici 2020), I consider a decentralized monitoring structure thatis immune to collusion but show that learning
environments that are subject to information
attrition also require ethical behavior.
I. ModelA. Investigation Structure
We consider a sequence ​​​{​Mn​ ​​}​​n∈ℕ​​​ of agents
forming a monitoring chain.
Round 0: Agent ​​M0​ ​​​ decides between committing a crime and abstaining from it.
Round 1: Agent ​​M​1​​​ decides between investigating ​​M​0​​​ at cost ​c > 0​and shirking at no(direct) cost. If ​​M​0​​​committed the crime and ​​M​1​​​
investigates him, ​​M​1​​​ discovers ​​M0​ ​​​’s guilt withprobability ​λ ∈ ​(0, 1]​​. If ​​M1​ ​​​ shirks, he finds
nothing.
Agent ​​M1​ ​​​then makes a public announcement: ​​M1​ ​​​can either claim that he discoveredthat ​​M​0​​​was guilty or say that he found nothing.
If ​​M​1​​​made no discovery, which happens
because ​​M0​ ​​​ was innocent, ​​M​1​​​ was unlucky(when λ​ < 1​, the investigation does not always
succeed even if M​​​ 0​​​ is guilty), or because M​​​ 1​​​
shirked, ​​M1​ ​​​can still claim that M​​​ 0​​​ committed acrime, that is, make a wrongful accusation. An
accusation is wrongful if it is unsubstantiated. It
does not imply, per se, that the accused is innocent of the crime.

* Northwestern University (email: b-strulovici@northwestern.edu). I am grateful to Steve Callander, Théo
Durandard, Ariel Rubinstein, Leeat Yariv, and participantsof 2019 Summer School of the Econometric Society in
Sapporo, Japan, and the ASSA 2021 Virtual Annual Meeting.
†
Go to https://doi.org/10.1257/pandp.20211053 to visit
the article page for additional materials and author disclosure statement.
1
See also Levine and Modica (2016). Rahman (2012)
proposes an ingenious way to incentivize ­first-degree monitoring, although this way is vulnerable to corruption between
the principal and the monitored agent.
2 ­Well-known examples in the United States include thefindings of the Knapp and Mollen commissions in New
York City. Most countries exhibit significantly more corruption. See, e.g., Transparency International: https://www.
transparency.org/en/cpi.
544

VOL. 111LEARNING AND CORRUPTION ON MONITORING CHAINS

If ​​M1​ ​​​announces that he found nothing, thegame ends.3 Otherwise, it moves to round 2, andanother agent, ​​M​2​​​, is tasked to investigate M​​​ 1​​​’sclaim that ​​M​0​​​ committed a crime.
Round 2: ​​M​2​​​ decides between investigating ​​M1​ ​​​ at cost ​c​and shirking. If ​​M​1​​​ accused ​​M0​ ​​​without proof and ​​M​2​​​ investigates him, ​​M2​ ​​​ discovers ​​M​1​​​’s wrongdoing with probability ​λ​.
Agent ​​M2​ ​​​then makes a public announcement: ​​M2​ ​​​can either claim that he discoveredthat ​​M​1​​​ wrongfully accused ​​M0​ ​​​ or announce that
he discovered nothing against M​​​ 1​​​. If ​​M​2​​​ madeno discovery (which can arise if M​​​ 1​​​ is innocent, ​​M2​ ​​​ was unlucky, or ​​M2​ ​​​ shirked), ​​M​2​​​ canstill accuse ​​M​1​​​of making an unsubstantiatedclaim.
If ​​M​2​​​announces that he has found nothing
against ​​M1​ ​​​, the game ends. Otherwise, the gamemoves to round 3. Another agent, ​​M​3​​​, is tasked
with investigating ​​M​2​​​’s claim, and so on.
Collusion: If ​​M​n​​​ discovers wrongdoingby ​​M​n−1​​​, he can offer to hide his discovery
in exchange for a transfer ​​τ​n​​​ from ​​Mn−1​ ​​​. Thispossibility captures a local form of collusion
between ​​M​n​​​ and ​​Mn−1​ ​​​. While we focus on a
take-it-or-leave-it offer by M​­​​ n​​​, other efficientbargaining protocols, in which M​​​ n−1​​​ has somebargaining power, would yield similar results.4
B. Payoffs
If ​​M​n​​​claims to have discovered that ​​M​n−1​​​
has made a wrongful accusation, he receivesa monetary reward ​​R​n​​  ≥ 0​for his discovery,and ​​Mn−1​ ​​​ receives a punishment ​​P​n−1​​  ≥ 0​. By
making this claim, ​​M​n​​​exposes himself to therisk of being accused (rightly or not) of wrongful behavior.
Players are assumed to be risk neutral,
and all payoffs (investigation costs, rewards,
3
This assumption drastically simplifies the analysis, as it
implies that there is only one public history at the beginningof any round that is reached (see Section II). The analysis
can be extended to the case in which a monitor who finds
nothing can also be investigated. Theorem 1 continues tohold under the additional assumption that λ​ < 1 / 2​. See
Section III.
4
If two of more agents were used at each level of themonitoring chain, this may make collusion harder, although
it would also create a collective decision problem for thefindings.

545p­ unishments, and transfers) are additively separable. A monitor’s payoff if the game ends
before his round is normalized to 0.
It is simplest to think of the punishment ​​P​n−1​​​
as a fine, but ​​P​n−1​​​could alternatively represent a­nonmonetary, additively separable punishment.
In principle, we could allow ​​M​n−1​​​ to be exonerated if M​​​ n​​​’s accusation is claimed (rightly ornot) to be unsubstantiated by ​​M​n+1​​​, punishedagain if ​​M​n​​​’s accuser is found to have lied, andso on. We rule this out for simplicity and assume
that ​​Mn−1​ ​​​is punished whenever he is accused.5
We start with the following observation.
Let ​​α​n+1​​​denote the probability that ​​M​n+1​​​
wrongfully accuses ​​M​n​​​ conditional on reachinground ​n + 1​.
LEMMA 1: Suppose that ​​Mn​ ​​​ discovers that ​​M​n−1​​​
made a wrongful claim. Then, ​​M​n​​​ surely hides
his discovery unless(1)​​Rn​ ​​  − ​α​n+1​​ ​Pn​ ​​  ≥ ​Pn−1​ ​​.​PROOF:
Suppose that M​​​ n​​​ discovers that ​​M​n−1​​​ hasmade a wrongful claim. If ​​M​n​​​ makes his (rightful) discovery public, his payoff increases by
an immediate reward ​​R​n​​​, less an expected fine
if he is wrongfully accused by ​​M​n+1​​​, whichhappens with probability α​​​n+1​​​ and carries
an expected cost of ​​P​n​​​ whenever it occurs,
while ​​M​n−1​​​ receives the fine ​​P​n−1​​​. Therefore,
revealing the discovery results in a joint surplus to M​​​ n​​​ and ​​M​n−1​​​ of ​​R​n​​  − ​α​n+1​​ ​Pn​ ​​  − ​P​n−1​​​
relative to the pair’s surplus if ​​M​n​​​ hides thediscovery, thereby ending the game. Sincetransfers allow M​​​ n​​​ and ​​M​n−1​​​ to maximize theirjoint surplus, ​​M​n​​​produces the evidence inequilibrium only if R​​​ n​​  − ​α​n+1​​ ​Pn​ ​​  − ​P​n−1​​  ≥ 0​.
Otherwise, ​​Mn​ ​​​hides the evidence in exchange
for a transfer ​​τ​n​​  ∈ ​
[​Rn​ ​​  − ​α​n+1​​ ​Pn​ ​​, ​P​n−1​​]​​. Undera ­take-it-or-leave-it offer protocol, M​​​ n​​​ gets atransfer ​​τn​ ​​  = ​Pn−1​ ​​​if he gets to make the offer
and a transfer ​​τ​n​​  = ​Rn​ ​​  − ​α​n+1​​ ​Pn​ ​​​ if ​​Mn−1​ ​​​ gets tomake the offer. When both players have bargaining power, any transfer τ​​​ n​​​ between these bounds
can be microfounded by an efficient bargaining
protocol. ∎5A simple interpretation of this is that each agent lives
for two periods: monitoring in the first period and dying inthe second one.

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Thus, collusion can be prevented only
if ​​Rn​ ​​  ≥ ​Pn−1​ ​​  + ​α​n+1​​ ​Pn​ ​​​. When this conditionis violated, any public accusation made by ​​M​n​​​must be a lie, because true discoveries by M​​​ n​​​ arealways hidden. To avoid this, we focus on thecase in which ​​R​n​​  ≥ ​Pn−1​ ​​  + ​α​n+1​​ ​Pn​ ​​​ and assume
that agents do not collude (alternatively, we
could assume a strict inequality, which implies
that collusion does not occur). We call suchan equilibrium a ­collusion-free equilibrium.
Focusing on c­ ollusion-free equilibria simplifies
the analysis since monitors’ choices are reduced
to four possible strategies, described in the next
section.
II. Main Result

In this game, there is a unique relevant public history: conditional on reaching round n​ ​, allmonitors have claimed (rightly or not) that pastmonitors were lying.
In particular, there is a unique sequence of
rewards {​​​ ​Rn​ }​​ n≥1​​ ​​​ along this history.
Let ​​β​n​​  ∈ [​ 0, 1]​​denote the probability that ​​Mn​ ​​​investigates ​​M​n−1​​​and truthfully announces his
finding, conditional on reaching round n​​.
If ​​βn​ ​​  = 0​, this means that ​​M​n​​​ either ends thesequence without performing an investigationor that he accuses ​​M​n−1​​​ regardless of ​​Mn−1​ ​​​’sactual guilt. Either way, ​​M​n​​​’s announcement is
uninformative about ​​M​n−1​​​’s actions whenever​​βn​ ​​  = 0​.
THEOREM 1: Suppose that λ​ < 1​ (imperfectsignal). If the sequence ​​​{​R​n​​}​​n≥1​​​ is bounded
above, then ​​β​n​​  = 0​for all ​n​in all ­collusion-freeequilibria.

Theorem 1 shows that truthful investigations
are impossible unless an unbounded amount of
resources can be devoted to monitoring: regardless of the budget available for monitoring, there
is a path of observations, which has positive
probability, that will exceed this budget. In particular, sequential monitoring that withstands a
local form of collusion is impossible in any society with bounded resources.
To prove this result, we begin by expressing ​​Mn​ ​​​’s payoffs for each possible strategy.
Let ​​γn−1​ ​​​denote the probability that M​​​ n−1​​​ hasmade a wrongful claim, conditional on round n​ ​being reached, and δ​​​n+1​​​ denote the probability that M​​​ n+1​​​ accuses ​​M​n​​​ with probability 1­regardless of ​​Mn​ ​​​’s guilt, conditional onround ​n + 1​being reached.6
Agent ​​M​n​​​has four pure strategies (which hemay, and typically will, randomize over).
Truthful Investigation.—​​Mn​ ​​​ investigates ​​M​n−1​​​and reports his finding truthfully. The expected
payoff of this strategy is(2)​λ ​γn−1​ ​​​(​R​n​​  − ​δ​n+1​​ ​Pn​ ​​)​  − c.​Indeed, ​​Mn​ ​​​incurs the investigation cost c​ ​ and,with probability ​λ ​γn−1​ ​​​, discovers that ​​M​n−1​​​
made a wrongful claim, resulting in expected
reward ​​R​n​​​. However, with probability ​​δn+1​ ​​​, ​​M​n​​​
is falsely accused by ​​M​n+1​​​of making a wrongfulclaim.7
Blind Accusation.—​​Mn​ ​​​ shirks and accuses​​M​n−1​​​. The resulting expected payoff is(3)​​Rn​ ​​  − ​(λ ​βn+1​ ​​  + ​δ​n+1​​)​ ​Pn​ ​​.​

Indeed, with probability λ​ ​βn+1​ ​​​, ​​Mn+1​ ​​​ investigates truthfully ​​M​n​​​and discovers his wrongful behavior, and with probability ​​δ​n+1​​​, ​​M​n+1​​​
accuses ​​Mn​ ​​​ with probability 1 (possibly after
shirking or after investigating ​​M​n​​​).
“Switching.”—​​M​n​​​ investigates ​​Mn−1​ ​​​ butaccuses ​​Mn−1​ ​​​even if he doesn’t find anything.
The resulting payoff is(4) ​λ ​γn−1​ ​​​(​Rn​ ​​  − ​δ​n+1​​ ​Pn​ ​​)​  − c+ ​(1 − λ ​γ​n−1​​)​​(​Rn​ ​​  − ​(λ ​βn+1​ ​​  + ​δ​n+1​​)​  ​Pn​ ​​)​.​

Indeed, with probability ​λ ​γ​n−1​​​, ​​Mn​ ​​​ discovers
that ​​M​n−1​​​lied and rightfully accuses him. With
probability ​1 − λ ​γ​n−1​​​, ​​M​n​​​ discovers nothing but
makes up an accusation, at the risk of getting
himself accused (rightfully or not).

6Note that ​αn​is smaller than ​γ​n​, because the former is ex
ante at round ​n​and the latter is conditional on round ​n + 1​
being reached.
7
Since ​M​n​was truthful, he will be falsely accused if ​M​n+1​​
uses either the “switching” or the “blind accusation” strategy, which has probability ​δn​ +1​​.

VOL. 111LEARNING AND CORRUPTION ON MONITORING CHAINS

Closing.—​​Mn​ ​​​can save the cost of an investigation and simply end the game by announcingthat he found nothing, in which case the payoff
is 0.
Remark: ​​δ​n​​​is the probability that M​​​ n​​​ chooses
either the “blind accusation” or the “switchingstrategy,” ​​β​n​​​is the probability that ​​M​n​​​ chooses thetruthful investigation strategy, and 1​ − ​β​n​​  − ​δ​n​​​is the probability that ​​M​n​​​ chooses the “closing”
strategy.

LEMMA: 2 If ​​Rn​ ​​  − ​(λ ​βn+1​ ​​  + ​δ​n+1​​)​ ​P​n​​  > 0​, ​​Mn​ ​​​always accuses ​​Mn−1​ ​​​, i.e., ​​δn​ ​​  = 1​ and ​​βn​ ​​  = 0​.
PROOF:
From payoff equations (2)–(4) and the zero
payoff from closing the case, the lemma’s
inequality implies that blindly accusing M​​​ n−1​​​
dominates closing the case and that switching
dominates truthfully investigating ​​M​n−1​​​, which
proves the lemma. ∎LEMMA 3: If ​​β​n​​  = 0​, then ​​β​k​​  = 0​for all​k ∈ ​
1,
…  
,
n
−
1
.
​​
{
}
PROOF:
If ​​β​n​​  = 0​, it is optimal for ​​M​n−1​​​ to end thegame or to blindly accuse ​​M​n−2​​​ at no cost:if ​​M​n−1​​​ makes an accusation, ​​M​n​​​ will either
close the investigation or accuse M​​​ n−1​​​ regardless of the veracity of M​​​ n−1​​​’s claim. Eitherway, ​​β​n−1​​  = 0​. The result follows by backward
induction. ∎LEMMA 4: If ​​β​n+1​​  < c/​Pn​ ​​​, then ​​β​n​​  = 0​.

PROOF:
From payoff equations (2) and (3), ​​Mn​ ​​​ strictlyprefers blindly accusing ​​M​n−1​​​ over investigating ​​Mn−1​ ​​​ truthfully unless  ​
λ ​γ​n−1​​​(​Rn​ ​​  − ​δ​n+1​​ ​Pn​ ​​)​  − c547

PROOF OF THEOREM 1:
From Lemmas 2 and 3, either there
exists a round ​​N​1​​​ beyond which ​​R​n​​  − ​​ ​​  + ​δ​n+1​​)​ ​Pn​ ​​  > 0​for all ​n > N​ or ​​βn​ ​​  = 0​(λ ​βn+1
for all ​n ≥ 1​. From (1), ​​Rn+1​ ​​  ≥ ​P​n​​​ for all ​n​.
This implies that truthful investigation is possible only if for all ​n ≥ ​N1​ ​​​,

	​​R​n​​  ≤ ​(λ ​βn+1​ ​​  + ​δ​n+1​​)​ ​Rn+1​ ​​.​

Likewise, from Lemmas 3 and 4,
either there exists a round N​​​2​​​ beyond
which ​​βn+1​ ​​  ≥ c/​Pn​ ​​​ for all ​n ≥ ​N​2​​​ or ​​β​n​​  = 0​
for all ​n ≥ 1​. From (1), ​​Rn+1​ ​​  ≥ ​P​n​​​ for all ​n​.
Therefore, truthful investigation is possible onlyif	​​β​n+1​​  ≥ c/​Rn+1​ ​​​for all ​n ≥ ​N​2​​​. In particular, this implies
that ​​Rn​ ​​ ≥ c​for all ​n ≥ ​N2​ ​​  + 1​.
Now suppose that {​​ ​Rn​ }​​ ​​is bounded above
by some constant ​​R̄ ​​ (which, as noted, must
exceed ​c​). Combining previous observations
and letting ​N = max​{​N1​ ​​, ​N2​ ​​}​  + 1​, truthfulinvestigation is possible only if• ​​β​n​​  ≥ c/​R̄ ​​,1• ​​ _ ​ ​R​ ​​  ≤ ​Rn+1​ ​​​,λ ​β​ ​​  + ​δ​ ​​ n
n+1

n+1

for all ​n ≥ N​. Since ​​δn+1​ ​​  ≤ 1 − ​β​n+1​​​ and thefunction ​x ↦ λx + ​(1 − x)​​ is decreasing on​​[0, 1]​​ for ​λ ≤ 1​, this implies that(5)​​R​n+1​​  ≥ G ​Rn​ ​​​for all ​n ≥ N​, where ​G = 1/​(λ(c/​R ​̄) +(1 − c/​R̄ ​))​​is strictly greater than 1 because​λ < 1​. Iterating (5) from ​N​to ​N + T − 1​, weget    ≥ ​R​n​​  − ​(λ ​βn+1​ ​​  + ​δ​n+1​​)​ ​Pn​ ​​,​

	​​R​N+T​​  ≥ ​G​​  T​ ​R​N​​​,

which is possible only if

and hence ​​R​N+T​​​ diverges to ​+ ∞​as ​T → + ∞​,which contradicts the assumption that {​​ ​Rn​ ​​}​​ wasbounded above.
This shows that either {​​​ ​Rn​ }​​ n≥1​​ ​​​ is unboundedor ​​βn​ ​​  = 0​for all ​n ≥ 1​, as claimed by the­theorem. ∎​​βn+1​ ​​  λ ​Pn​ ​​  ≥ c + ​(1 − λ ​γ​n−1​​)​​(​R​n​​  − ​δ​n+1​​ ​P​n​​)​.​The last term is nonnegative, and ​λ ≤ 1​. The
claim follows. ∎548

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III. Extension

This section sketches a simple extension of theanalysis, in which ​​M​n​​​is investigated even if hedoes not accuse M​​​ n−1​​​. Monitors no longer havethe ability to end the game, and a public history
now consists of a sequence of declarations, one
for each round n​ ​, in which M​​ n​ ​​​ either accuses ​​M​n−1​​​(denoted “​a​”) or doesn’t (denoted “​d​”).
The rewards, punishment, and strategies candepend arbitrarily on past history. To capture
this, we modify earlier notation as follows.
Fixing a history up to round n​ ​, let ​​P​  an​​​  denote ​​Mn​ ​​​’spunishment in round ​n + 1​if he accused ​​M​n−1​​​in round ​n​ and ​​Mn+1​ ​​​claims that this accusation was wrongful and ​​P​  dn​​​  denote ​​M​n​​​’s punishment if he did not accuse M​​​ n−1​​​ and ​​Mn+1​ ​​​ claims
that ​​M​n​​​was either shirking or corrupt. For simplicity, we do not distinguish between these twowrongful behaviors when ​​M​n​​​ made no accusation. Let β​ ​​ an+1​​​denote the probability that M​​​ n+1​​​
truthfully investigates ​​M​n​​​ conditional on ​​M​n​​​accusing ​​Mn−1​ ​​​ and ​​β​  dn+1​​​denote the same probability conditional on ​​M​n​​​ not accusing ​​M​n−1​​​.
Let ​​δ​  an+1​​​denote the probability that M​​​ n+1​​​ surely
accuses ​​Mn​ ​​​without investigating him conditional on M​​​ n​​​ accusing ​​M​n−1​​​ and ​​δ​  dn+1​​​denote thesame probability conditional on M​​​ n​​​ not accusing ​​M​n−1​​​. Finally, let ​​λ​​  a​​denote the probability that ​​M​n+1​​​ discovers ​​Mn​ ​​​ wrongdoing
if he investigates ​​M​n​​​ and ​​M​n​​​ wrongfully
accused ​​M​n−1​​​ and ​​λ​​  d​​denote the probability
that ​​Mn+1​ ​​​ discovers wrongdoing by ​​Mn​ ​​​ if he investigates ​​Mn​ ​​​ and ​​Mn​ ​​​shirked or hid evidence against​​Mn−1​ ​​​.
The ­collusion-free condition in round ​n​becomes​​R​n​​  − ​δ​  an+1​​ ​P​  an​ ​  − ​P​n−1​​  ≥ − ​(​δ​  dn+1​​  + ​λ​​  d​ ​β​  dn+1)​ ​ ​P​  dn​,​  ​and ​​Mn​ ​​​now prefers truthfully investigating​​ n−1M​ ​​​over the “switching” strategy (defined previously) if​​R​n​​  − ​(​δ​  an+1​​  + ​λ​​  a​ ​β​  an+1​)​ ​P​  an​ ​  ≤ − ​δ​  dn+1​​ ​P​  dn​.​  ​Isolating ​x = ​R​n​​  + ​δ​  dn+1​​ ​P​  dn​ ​  − ​δ​  an+1​​ ​P​  an​​​  in the last
two inequalities yields​​P​n−1​​  − ​λ​​  d​ ​β​  dn+1​​ ​P​  dn​ ​  ≤ x ≤ ​λ​​  a​ ​β​  an+1​​ ​P​  an​,​  ​which implies that​​P​n−1​​  ≤ ​λ​​  d​ ​β​  dn+1​​ ​P​  dn​ ​  + ​λ​​  a​ ​β​  an+1​​ ​P​  an​.​  ​It follows that if ​​λ​​  d​  + ​λ​​  a​  < 1​, there must exista path along which the punishment sequence isunbounded.
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